Geodesic flows on moduli spaces, continued fractions and Lyapunov exponents

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Matematica

Anno Accademico 2006 – 2007

Tesi di Laurea Specialistica:

Geodesic flows on moduli spaces,

continued fractions and Lyapunov

exponents

Candidato:

Dario Beraldo

Relatore:

Prof. Stefano Marmi

Controrelatore:

Prof. Marco Abate

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Preface

It is known that there is a simple and beautiful relation between continued fractions and geodesics in the hyperbolic upper half-planeH. An amazing and very geometrical explanation goes as follows. Consider the triangle in H with vertices at 0, 1 and∞ (this is a triple covering of the classical modular surface SL(2, Z)\H) and the tiling ofH generated by it via the integral translations and the inversion,

z_{7→ z ± n,} z_{7→ −}1 z.

Consider now a segment of geodesic (that is, a portion of a circumference) starting from a point iy in the vertical half-line outgoing from zero and arriving at x_{∈ (0, 1) on the real axis. The geodesic hits a sequence (possibly finite) of triangles} of the tiling, and we code it with the rule given by the figure.

In more precise terms, if the geodesic crosses the triangle leaving only one vertex to the left, we write down L; if it crosses the triangle leaving only one vertex to the right, we write R. We end up with a sequence of L and R of the type

Rn1_{, L}n2_{, R}n3_{, . . . ,}

where we grouped together the maximum number of contiguous symbols corre-sponding to the same letter. Then, it turns out that the natural numbers [nj] form

the continued fraction expansion of x: x = J0, n1, n2, n3, . . .K =

1 n1+_n 1

2+_n3+···1

.

Another nice representation of the geodesic flow on H derives from the follow-ing considerations. The unit tangent bundle of H is nothing but PSL(2, R): this

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identification is obtained via Mobius transformations. For instance, a vertical unit vector at the point et

· i is given by the matrix

et/2 ₀

0 e−t/2

. This means that the map

t7→

et/2 ₀

0 e−t/2

parametrizes the vertical geodesic of speed 1 starting from i. This simple case allows to show that the geodesic flow is given by right multiplication of the diagonal subgroup Et₌ et/2 ₀ 0 e−t/2 : t_{∈ R} ; i.e. Gt: PSL(2,R) → PSL(2, R), Gt(M )_{7→ M · E}t.

Furthermore, any element of PSL(2,R) is obviously seen as a torus of area 1; then the flow acts dilating the torus along the horizontal direction and shrinking it along the vertical one, preserving the area.

It is more interesting to consider the action of this flow on the space of equivalence classes of tori, that is, the quotient map

Gt: SL(2,Z)\ SL(2, R) → SL(2, Z)\ SL(2, R), Gt(M )_{7→ M · E}t.

This leads to a simple example of a general procedure, called renormalization dy-namics, that we now describe. Assume we are given a flow Ft_{on a space X that}

descends to a flow on a quotient space G_{\X, where G is a group. Let D be a} fundamental domain for the action of G on X and choose x_{∈ D: its image ˜x under} the flow may eventually get out of_{D but it can be always brought back to D by an} element of A(x)_{∈ G, i.e.}

f (x)_{≡ A(x) · ˜x ∈ D.} The main hope is that the sequence

A(x), A(f (x)), . . . , A(fn_{(x)), . . .}

gives non-trivial information on the nature of the point x, on the dynamical prop-erties of the flow, and so on. This procedure has the great advantage to discretize the time; still, one should exclude orbits that remain inside D forever. Typically this is not at all a crude restriction.

To implement this method in the case of the geodesic flow on SL(2,_{Z)\H, we} neglect tori whose lattices have either horizontal or vertical vectors; they correspond to geodesics that go to infinity and so they are not of interest. A fundamental domain_{X in SL(2, R) for the action of SL(2, Z) is given by tori that are realized as} a suspension over a rotation; precisely

a b c d

∈ X

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PREFACE v

if and only if either a > 1 > c > 0, d > _{−b > 0 or c > 1 > a > 0, −b > d > 0.} The dynamics induced on R by the renormalization map is very much equivalent to the Gauss map x_{7→ {1/x} (the basic step of the continued fraction algorithm).} In fact, starting from (x, 1) with x_{∈ (0, 1), the flow acts giving (1, 1/x) and then,} by renormalization, we get (1,_{{1/x}). Iterating, we obtain}

(x, 1)_7→ 1, 1 x 7→ ( 1 1 x ) , 1 ! 7→ · · · .

At this point, we are naturally tempted to extend all these phenomenona. There are, at least, two possibilities, namely

• consider SL(n, Z) and SL(n, R), with n > 2;

• observing that tori are compact Riemann surfaces of genus 1, consider the case of genus g.

We shall choose the second option. As in the case of genus 1, up to a negligible set, every surface of genus g is realized as a suspension over an interval map of special type, called interval exchange transformation. See the figure.

Each of these surfaces is called a translation surfaces, since it is realized as planar polygon in the complex plane (with an even number of sides pairwise identified), and the flat metric onC descends to a flat metric on each surface (outside a finite set of singularities).

An interval exchange transformation (IET) is a map that breaks an interval into a finite number d of subintervals and then rearranges them in a different order. For

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instance, if d = 2 then there are essentially two cases: the first is the trivial case (the identity map); the second is ”morally” a rotation. For this reason we sometimes say that interval exchange transformations are a generalization of rotations. Interval exchange transformations are very easy to define (as we have just seen), but, in spite of their simplicity they present a quite rich dynamics. This richness comes from the combinatorics of the IET (that is, how it permutes the subintevals) and also from the relationship between the lengths of the subintervals.

We can extend verbatim (in reality, up to a factor 2) the geodesic flow to a flow on the space of translation surfaces: the horizontal lengths are dilated of a factor et_{, while the vertical directions are contracted, by the same factor.}

The renormalization map also admits a generalization, the so-called (invertible) Rauzy-Veech renormalization map. Roughly speaking, the Rauzy-Veech renormal-ization map acts cutting the triangle given by the two rightmost edges of the surface, and then pasting it back on the surface using the identifications between the edges. Actually, there are two possibilities for pasting, as the figure explains.

In the case of genus g, this map corresponds to the Farey map, which is nothing but a slow version of the continued fraction algorithm. The Gauss map corresponds to the Zorich renormalization map Z, which is just obtained by grouping a finite number of Rauzy-Veech renormalizations of the same ”type”.

From an ergodic theoretic point of view, the analogy between the simple case g = 1 and the general case g > 1 is also preserved. Indeed, it is well-known that the Gauss probability

µG(E) = 1 log 2 Z E 1 1 + xdx,

which is equivalent to the Lebesgue measure on (0, 1), is invariant under the Gauss map. Analogously, there exists a probability, the Zorich measure µZ, on the space

of IETs that is ergodic and invariant under Z. Moreover, µZ is equivalent to the

Lebesgue measure and it is the unique probability with these properties.

The action of the Zorich renormalization map is the projectivization of a linear isomorphism and this allows us to define a linear cocycle over Z, called the Zorich cocycle, FZ. One checks that FZ satisfies the hypothesis of the multiplicative

ergodic theorem, and so it possesses a well-defined Lyapunov spectrum, (0.1) θ1>θ2>· · · > θ2g−1 >θ2g.

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PREFACE vii

To understand the meaning of the Lyapunov exponents θj, we look at the genus 1

case: we have that θ2=−θ1 and

θ1=

π2

12 log 2 = limn→∞

1 nlog qn,

where qn is the denominator of the nth convergent to the continued fraction of a

typical x_{∈ R.}

It was conjectured by M. Kontsevich and A. Zorich (1995) that the inequalities are in (0.1) are strict, that is,

θ1> θ2>· · · > θ2g−1 > θ2g.

This conjecture is true and it was proved by A. Avila and M. Viana in 2005. Structure of the thesis. In the first chapter we shall study the geodesic flow on the modular surface and its links with continued fractions. In the second and third chapter, we introduce interval exchange transformations and translation surfaces, extending the definitions given in the first chapter (defining Rauzy-Veech and Zorich renormalization operators). Then, we will deal with ergodicity, proving that the Zorich renormalization admits and invariant ergodic probability. In the fifth and sixth chapters, we will define the Zorich cocycle and show that it has a well-defined Lyapunov spectrum. Some elementary properties of this spectrum are also derived. Finally, in the last chapters we state and prove a sufficient criterion for the simplicity of the spectrum and check that it applies to the Zorich cocycle.

Acknowledgements. I would like to express my thanks to Stefano Marmi who introduced me into this subject, to Artur Avila and Marcelo Viana for their kindness and helpfulness at IMPA, to Marco Abate for his useful observations.

I am also grateful to Nelita, Priscilla (as minhas irm˜as mais velhas), Luciana, Barbara, Celina and Ma´ıra. Finally, I thank my family and my friends, especially Alberto and Nicola.

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CHAPTER 1

Geodesic flow on the hyperbolic plane

This chapter has an introductory character and serves as a model and mo-tivation for the following chapters. It mainly deals with continued fractions and geodesics in the hyperbolic half-plane. These objects are reciprocally connected, and we explain in full detail the links between them.

First of all, it turns out that the geodesic flow on the unit tangent bundle of the hyperbolic upper half-plane H can be represented as the action (by right multiplication) of the diagonal subgroup

et/2 ₀ 0 e−t/2 : t∈ R

on PSL(2,R). Any element of PSL(2, R) represents a torus of area 1 and so S = PSL(2, Z)\ PSL(2, R)

is the space of lattices of unitary area. We shall study the flow induced on it by the geodesic flow on H. Choosing a particular fundamental domain in H for M1= PSL(2,Z)\H, we will show that the geodesic flow on S is conjugated to the

Gauss map, that constitutes the basic step of the continued fraction algorithm. 1. Continued fractions

A finite continued fraction Ja0, . . . , aNK is an expression of the following form:

Ja0, . . . , aNK = a0+ 1 a1+_a 1 2+ 1 ...+ 1 aN ,

provided that the quotient makes sense. More precisely, Ja0, . . . , aNK can be defined

recursively as follows: set Ja0K = a0 and, for n 6 N ,

Ja0, . . . , anK = a0+

1 Ja1, . . . , anK

. From now on we assume that

a0∈ Z and aj ∈ N\ {0} , for j > 1.

This implies that Ja0, . . . , aNK is well-defined and rational. It is easy to find p, q

coprimes such that Ja0, . . . , aNK = p/q.

Proposition _{1.1. Let Ja}₀_{, . . . , a}_N_{K. Set p}₋₁ _{= 1, q}₋₁ _{= 0, p}₀ _{= a}₀_{, q}₀ _{= 1} and, for 1 6 j < N , pj+1= aj+1pj+ pj−1; qj+1= aj+1qj+ qj−1. Then (1.1) Ja0, . . . , ajK = pj qj 1

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and

(1.2) pjqj−1− pj−1qj = (−1)j−1.

Observe that (1.2) implies that pj and qj are always coprime. The fraction p_qn_n

is called the nth_{convergent to the continued fraction.}

Observation _{1.2. Clearly, q}_n>2qn−2. It follows that qn>2(n−1)/2.

Proof of Proposition 1.1. _{It is a standard induction, using the obvious} fact: Ja0, . . . , ak, ak+1K = s a0, . . . , ak+ 1 ak+1 { , for all 0 6 k 6 N_{− 1.}

The second equality is also a straightforward induction. Consider the following map, called the Gauss map:

G : [0, 1)→ [0, 1), G(x) = 0 if x = 0 1 x otherwise. Let also T (x) = [1/x], for x_{6= 0.}

The continued fraction algorithm is defined as follows: given x ∈ (0, 1), set x1= G(x) and a1= T (x). We have

x = 1 a1+ x1

;

if x1 = 0 the algorithm stops giving the result x = J0, a1K, otherwise we let x2 =

G(x1) = G2(x) and a2= T (x1) = T (G(x)). Hence,

x = 1

a1+_a₂_+x1 ₂

.

As before, if x2 = 0, the algorithm is finished and the result is x = J0, a1, a2K;

otherwise we continue defining x3= G3(x) and a3= T (G2(x)).

The general procedure is clear: we define xk = Gk(x) and ak = T (Gk−1(x)),

provided that Gk−1_(x)

6= 0. If Gk−1_{(x) = 0, then the algorithm stops and}

x = J0, a1, . . . , ak−1K .

If we want to start with x∈ R instead of x ∈ (0, 1), then it suffices to take a0= [x]

and apply the algorithm to_{{x}. The only change is that, at each step, we obtain} Ja0, a1, . . . , ak−1K instead of J0, a1, . . . , ak−1K.

It is natural to ask for which x the algorithm stops and whether it gives a unique expansion of x in continued fraction.

Proposition _{1.3. The algorithm stops if and only if x}_{∈ Q. Moreover, each} rational number has exactly two continued fraction expansions, one of the form Ja0, . . . , akK with ak > 1 and the other one with Ja0, . . . , ak− 1, 1K.

Proof. _{It suffices to prove the claims for x}_{∈ Q ∩ (0, 1). Let x = d}₁_/d₀_{, with} 0 < d1< d0. If G(x) > 0, then G(x) = d0 d1 =d2 d1 ,

with 0 < d2 < d1 since G(x) ∈ (0, 1). If Gk−1(x) 6= 0, the argument can be

repreated, yielding a sequence

d0> d1>· · · > dk> 0.

This implies that some iterate of x under G is zero, and the continued fraction algorithm gives a finite continued fraction for x. Viceversa, if x = Ja0, . . . , aNK,

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1. CONTINUED FRACTIONS 3

Clearly, Ja0, . . . , aNK = Ja0, . . . , aN − 1, 1K when aN > 1. Assume that

x = Ja0, . . . , aNK = Jb0, . . . , bMK ,

with aN > 1 and bM > 1. We are left to prove that M = N and that aj = bj for

all 0 6 j 6 N . First of all, [x] = a0= b0. Then

G(x) = Ja1, . . . , aNK = Jb1, . . . , bMK .

Repeating the procedure, we end up with

aN = JaNK = JbN, . . . , bMK

and this is absurd, unless N = M and aN = bN.

The next lemma suggests there is some number theoretical connection between continued fractions and the Lie group SL(2,Z).

Lemma _{1.4. If A =}

a b c d

∈ SL(2, Z) with c > d > 0, then there ex-ists a continued fraction Ja0, . . . , aNK such that a/c = Ja0, . . . , aNK and b/d =

Ja0, . . . , aN −1K.

Proof. _{We can write a/c = Ja}₀_{, . . . , a}_N_{K with N odd. Let p}_j_/q_j _{be the} convergents to this continued fractions as in Proposition 1.1. Since c > 0 and (a, c) = 1 (A _{∈ SL(2, Z)) we obtain that p}N = a and qN = c. Recall that

(₋₁₎(N −1)_{= p}

NqN −1− qNpN −1= ad− bc = dpN− bqN or

pN(d− qN −1) = qN(b− pN −1).

As (pN −1, qN −1) = 1, qN|(d − qN −1). On the other hand,|qN −1− d| < qN (because

qN >qN −1> 0 and qN = c > d > 0 by hypothesis) and this implies that qN −1= d.

Consequently, pN −1= b. Observe that 1 a1 0 1 · 1 0 a0 1 = a0a1+ 1 a1 a0 1 = p1 q1 p0 q0 and 1 0 a2 1 · 1 a1 0 1 · 1 0 a0 1 = p1 q1 p2 q2 . By induction, one proves that

(1.3) 1 an 0 1 · · · 1 0 a0 1 = pn qn pn−1 qn−1 if n is odd, and (1.4) 1 an 0 1 · · · 1 0 a0 1 = pn−1 qn−1 pn qn if n is even.

The group PSL(2,Z) is the quotient of SL(2, Z) by {I, −I}. Corollary _{1.5. Let} L = 1 0 1 1 and U = 1 1 0 1 . Then, PSL(2,Z) is generated by L and U.

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Proof. _{Preliminarly observe that} S = 0 −1 1 0 = U−1· L · U−1_, and so S is generated by U, L. Let A = a b c d

∈ SL(2, Z). We want to prove that A is generated by L, U . If c = 0, then either a = d = 1 or a = d = _{−1: in either case A = ±U}n_.

Analogously, if b = 0, we have A =_±Ln_{. So, we may assume that b}

· c 6= 0. Up to changing A with_{−A we may assume that c > 0. We have that}

A_· 1 _−p 0 1 = a b_{− pa} c d_{− pc} ,

so we may assume that 0 6 d < c. If d = 0, then c = 1 and b = −1, that is A = a −1 1 0 . Since Ua_{· S =} 1 a 0 1 · 0 ₋₁ 1 0 = a ₋₁ 1 0 = A,

we are done in this case. Finally, if 0 < d < c, we are in the case of Lemma 1.4: then we may assume that

A =

pN qN

pN −1 qN −1

and either Equation (1.3) or Equation (1.4) yields the conclusion (after

transpos-ing).

We are more interested in infinite continued fractions, defined by Ja0, . . . , an, . . .K = lim

n→∞Ja0, . . . , anK ,

provided that the limit exists. Indeed,

Proposition _{1.6. Given any a}₀ _{∈ Z and any sequence (a}_n₎_n>1 _{⊂ N\ {0},} the finite continued fractions Ja0, a1, . . . , anK do converge to a limit x ∈ R that

we denote Ja0, . . . , an, . . .K. Viceversa, the continued fraction algorithm applied to

x /_{∈ Q gives an infinite continued fraction (indeed the unique continued fraction)} whose value is x.

Proof. _{We know that the algorithm stops if and only if x}_{∈ Q. So, suppose} x /_{∈ Q ∩ (0, 1): the continued fraction algorithm gives that, for every n ∈ N,}

x = Ja0, . . . , an+ Gn(x)K .

An induction similar to that of Proposition 1.1 shows that x = pn+ G

n_(x)p n−1

qn+ Gn(x)qn−1

. Together with Equation (1.2), this yields

(1.5) x−p_qn

n =

(₋₁₎n_Gn_(x)

qn(qn+ Gn(x)qn−1);

consequently, the odd convergents fall to the right of x, while the even to the left of x. Since the recursive relation defining qn shows that qn → ∞ as n → ∞, we

immediately obtain that the convergents go to x; in other words, Ja0, . . . , anK = pn

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2. GAUSS MEASURE 5

We have proved that the continued fraction algorithm applied to x /_{∈ Q gives an} infinite continued fraction whose value is x.

Now, we want to prove that, given any sequence of positive natural numbers (a1, a2, . . .), the finite continued fractions J0, a1, . . . , anK converge to a limit. Indeed,

we have that pn qn − pn−1 qn−1 = (−1)n+1 qn−1qn and so J0, a1, . . . , aNK = N X n=1 pn qn − pn−1 qn−1 = N X n=1 (−1)n+1 qnqn+1 . Since qn→ ∞, we have that

J0, a1, . . . , an, . . .K = ∞ X n=1 (₋₁₎n+1 qnqn+1 <∞.

The uniqueness is now obvious, since the expansion Ja0, a1, . . . , an, . . .K is

ob-tained applying the continued fraction algorithm to x = Ja0, a1, . . . , an, . . .K.

Observation _{1.7. By Equation (1.5), we immediately deduce that}

(1.6) 1 qn(qn+ qn+1) < x₋pn qn < 1 qnqn+1 . 2. Gauss measure

The Gauss measure is defined by µG(E) = 1 log 2 Z E 1 1 + xdx for any measurable set E⊂ [0, 1].

Proposition _{1.8. The Gauss transformation G preserves the Gauss measure} µG.

Proof. _{It suffices to prove the statement for the intervals of the form (0, t).} We have that G−1((0, t)) = ∞ [ k=1 ₁ k + t, 1 k . Then, Z t 0 1 1 + xdx = ∞ X k=1 Z t/k t/(k+1) 1 1 + xdx = ∞ X k=1 Z 1/k 1/(k+t) 1 1 + xdx,

where the second equality is a straightforward calculation, and we are done. Consider the family of sets

∆(a1, . . . , an) =x ∈ [0, 1]\Q : T (Gk−1(x)) = ak, for 1 6 k 6 n

and the corresponding family of functions

ψa1,...,an: (0, 1)→ (0, 1), t 7→ J0, a1, . . . , an, tK .

Clearly, neglecting the set of rational numbers, ∆(a1, . . . , an) = ψa1,...,an((0, 1)),

and by a standard induction,

(1.7) ψa1,...,an(t) =

pn+ tpn−1

qn+ tqn−1

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Differentiating and using (1.2) we get that ψa1,...,an is increasing if n is even and

decreasing if n is odd. Consequently, ∆(a1, . . . , an) =    pn qn, pn+pn−1 qn+qn−1 if n is even; _p n+pn−1 qn+qn−1, pn qn if n is odd. Its Lebesgue measure is

(1.8) Leb(∆(a1, . . . , an)) =

1 qn(qn+ qn−1)

.

Theorem_{1.9. The Gauss measure is an ergodic probability for the Gauss map.} Proof. _{We fix a}₁_{, . . . , a}_n _{once and for all; to simplify the notation, let}

∆(a1, . . . , an) = ∆n and ψn= ψa1,...,an.

Suppose also that n is even, so that ψn is increasing. Trivially, if x ∈ ∆n, then

x = ψn(Gn(x)) (in fact ψn is an inverse branch of Gn). This implies that

s < Gn(x) < t _{⇐⇒ ψ}n(s) < x < ψn(t). and so (by (1.7)) Leb(∆n∩ T−n({s, t})) = ψn(t)− ψn(s) = t_{− s} (qn+ sqn−1)(qn+ tqn−1) . Then (Equation (1.8))

Leb(∆n∩ T−n({(} s, t))) = Leb(∆n) Leb((s, t))

qn(qn+ qn−1)

(qn+ sqn−1)(qn+ tqn−1)

. If follows that

(1.9) 1

2Leb(∆n) Leb((s, t)) 6 Leb(∆n∩ G

−n_{(s, t)) 6 2 Leb(∆}

n) Leb((s, t)).

Similar arguments yield (1.9) for n odd. This implies ergodicity. 3. L´evy constant

Proposition _{1.10. For almost every x}_{∈ R, the sequence (q}_n₎

n∈N is such that lim n→∞ log qn n = π2 12 log 2.

Proof. _{We may neglect the zero measure set of rationals. Given} x = Ja0, . . . , an, . . .K ,

denote by pj(x) and qj(x) the convergents to x; by definition we have that qj(G(x)) =

pj(x). Thus, the product n

Y

k=1

pn−k+1(Gk−1(x))

qn−k+1(Gk−1(x))

is equal to 1/qn(x); taking the logarithm, we have

(1.10) − log qn(x) = n X k=1 logpn−k+1(G k−1_(x)) qn−k+1(Gk−1(x)).

By Observation 1.2, and Equation (1.6), we have (for n > 1) x pn(x)/qn(x)− 1 6 1 qn+1(x) 62−n/2. The elementary inequality

| log(1 + s)| 6 4|s| if |s| 6 √1 2

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4. THE HYPERBOLIC PLANE 7 gives also log x_{− log}pn(x) qn(x) 64_{· 2}−n/2. In conclusion, by (1.10) n X k=1 log Gk−1(x)− log_q 1 n(x) 6 n X k=1 log G k−1_(x) − logpn−k+1(G k−1_(x)) qn−k+1(Gk−1(x)) and so n X k=1 log Gk−1(x)− log_q 1 n(x) 6 n X k=1 4· 2−n/2<∞. By the ergodic theorem, we have

1 n n X k=1 log Gk−1(x)_→ Z log dµG= 1 log 2 Z 1 0 log x 1 + xdx and so lim n→∞ 1 nlog qn(x) =− 1 log 2 Z 1 0 log x 1 + xdx = 1 log 2 Z 1 0 log(1 + x) x dx = 1 log 2 ∞ X k=0 (₋₁₎k (k + 1)2 = π2 12 log 2.

This concludes the proof.

4. The hyperbolic plane

As usual, we denote a complex number z as z = x + iy, with x and y real. The hyperbolic plane is the Riemann surface

H = {x + iy ∈ C : y > 0},

endowed with the natural complex structure inherited byC and with the Riemann-ian metric

dx2_{+ dy}2

y2 .

Denote by h·, ·i and k · k the euclidean scalar product and the euclidean norm. The definition means that the hyperbolic scalar product of two vectors v, w∈ TzH

equals tohv, wiH=hv, wi/y2: in particular

kvkH= kvk

|y|,

It follows that measures of angles coincide in the two metrics.

We shall denote by TH and T(1)_{H the tangent and unit tangent bundles of H,}

respectively. Thus, TH = [ z∈H {z} × C and T(1)H = [ z∈H, α∈[0,2π) {z} × {yeiα}.

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4.1. Mobius transformations. By definition, a Mobius transformation is a map T :CP1→ CP1such that

z_7→ az + b

cz + d, for a, b, c, d∈ C.

As one might trivially check, the set of Mobius transformations with complex co-efficients is a group, called Mob(C). We shall represent them as square matrices: given T ∈ Mob(C), let

AT = a b c d .

First observe that, det AT = 0 if and only if T is constant onCP1. Therefore, we

shall consider only non-constant transformations, i.e., those T such that det AT 6= 0:

they are all invertible. This is because the map T _{7→ A}T estabilishes a group

homo-morphism between GL(2,C) and Mob(C). The kernel of this map is the subgroup of {λI2 : λ∈ C∗}, thus Mob(C) ∼= PGL(2,C) = PSL(2, C). Recall that PSL(2, C)

is the group of matrix with unit determinant, with the antipodal identification A≈ −A.

In the same way, we can prove that Mob(R) ∼= PSL(2,R). Moreover,

Proposition _{1.11. The group of automorphisms of} _{H (i.e. the} biholomor-phisms φ : _{H → H) coincides with the group of orientation-preserving isometries} of H with respect to the hyperbolic metric. Moreover, this group is the group of non-constant Mobius transformations with real coefficients:

Aut(H) = Iso+_{(H) ∼}

= PSL(2,R).

Proposition _{1.12. The group PSL(2,}_{R) of isometries of H is generated by} horizontal traslations and inversion through the origin, i.e. by

Tx= 1 x 0 1 and S = 0 ₋₁ 1 0 .

Using the identification Mob(R) = PSL(2, R), we can obtain a matrix repre-sentation for H and its unit tangent bundle T(1)_H.

Let PSL(2,R) act on H in the obvious way:

a b c d

· z = az + b_{cz + d}.

We may lift this action to a natural action of PSL(2,R) on T(1)_{H: the relevant}

observation is that a Mobius transformation g acts on TH through its differential: g _{· (z, v) = (g(z), g}′_{(z)v). Moreover, by Proposition 1.11, lengths of vectors are}

preserved so that the action descends to T(1)_{H. In matrix notation,}

a b c d · (z, v) = az + b_{cz + d}, 1 (cz + d)2v .

Proposition _{1.13. The action of PSL(2,}_{R) on T}(1)_{H is transitive.}

Proof. _{It suffices to check that, for every (z, v)} _{∈ T}(1)_{H, there exists A ∈}

SL(2,R) such that A · (i, i) = (z, v). Observe that Rθ· (i, i) =

cos θ _{− sin θ} sin θ cos θ

· (i, i) = (i, eπ/2+2iθ₎

and Nx,y· (i, w) = √ y x/√y 0 1/√y · (i, w) = (x + iy, y · w)

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5. GEODESICS 9

for every θ, x, y _{∈ R, (y > 0), w ∈ C. In other words, every R}θ rotates the fiber

over i, and every Nx+iy just ”translates” the fiber over i to the one over z = x + iy.

So, if z = x + iy and v = eπ/2+2iθ_{, A = N}

x,y· Rθis as required.

One trivially checks that the stabilizer at (i, i) is {I, −I}, thus T(1)H ∼= PSL(2,R).

The action on H is therefore also transitive; the stabilizer at i is PSO(2, R), the subgroup of rotations. Consequently,

H ∼= PSL(2,R)/ PSO(2, R) = SL(2, R)/ SO(2, R).

In the sequel, we shall denote as M(z,v) the unique matrix (up to a sign) that

sends (i, i) to (z, v).

5. Geodesics Consider the curve γ(t) = et_{i, t}

∈ R. We claim that it is a geodesic for the hyperbolic metric. In fact, γ′_{(t) = e}t_{i and so, for every t,}

kγ′(t)_kH= kγ ′_(t)_k

Im (γ(t))= 1.

Moreover, if the geodesic starting from the point i with vertically upward unitary speed were not contained in_{{z : Re z = 0} then (by applying the isometry x+iy 7→} −x + iy) there would exist two geodesics with the same initial data, and this is a contradiction.

From γ we may obtain every geodesic in H: in fact, due to Propositions 1.11 and 1.13, there exists an isometry M (namely, Mz,v) that sends (i, i) to (z, v), so

that t_{7→ M · γ(t) is exactly the geodesic with initial data (z, v).}

Corollary _{1.14. The images of the geodesics of}_{H are vertical lines or} semi-circles centered on the real line.

Proof. _{By Proposition 1.12, it suffices to check that T}_x _{and S preserves the} set of vertical lines and semicircles centered on the real axis. This is obvious for Tx

and easy for S.

Definition_{1.15. The geodesic flow G =}_{Gt_}

t∈R is the flow on T(1)H defined

by Gt_{((z, v)) = g}

(z,v)(t), where g(z,v) is the unique geodesic at (z, v).

Noting that γ(t) = et_{i =} et/2 ₀ 0 e−t/2 , (and using Proposition 1.11) we obtain:

Gt((z, v)) = M(z,v)γ(t) = M(z,v)· et/2 ₀ 0 e−t/2 .

In conclusion, the geodesic flow on PSL(2,R) may be represented as the right action of_Tt₌ et/2 ₀ 0 e−t/2 .

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6. Modular surface

The geodesic flow just described admits another interpretation, which is indeed very important for us because it can be generalized (as we shall do in the following two chapters). A matrix A = a b c d

∈ SL(2, R) may be considered as a torus TA=R2/ (Zv1⊕ Zv2)

(or as a lattice Zv1⊕ Zv2) just defining v1 = (a, b)T and v2 = (c, d)T (changing

everything by a minus sign does not cause any problem). This torus has unit area, of course (i.e., the lattice has covolume one). The geodesic flow

Gt a b c d = et/2_{a e}−t/2_b et/2_c _e−t/2_d

acts dilating the torus along the horizontal direction and shrinking it along the vertical one, preserving the area.

A lattice L =Zv1⊕ Zv2has not a unique representation in terms of v1 and v2:

in fact, any matrix B _{∈ GL(2, Z) with det B = ±1 maps L onto L. If we specify} also an orientation (e.g. v1∧ v2= 1), then the group of linear maps preserving the

lattice is simply SL(2,Z).

It is therefore natural to study the space SL(2,Z)\ SL(2, R), considered as the unit tangent bundle

SL(2,_{Z)\ SL(2, R) ∼}= PSL(2,Z)\ PSL(2, R) = PSL(2, Z)\T(1)H of PSL(2, Z)_\H.

Definition _{1.16. The group Mod}₁ _{= PSL(2,}_{Z) is called modular group and} M1= PSL(2, Z)\H modular surface.

Let D ⊂ H the set D = {z : |z| > 1} ∩ {z : | Re(z)| 6 1/2}. It is well-known that D is a fundamental domain for the action of Mod1 onH. To be precise, let

T = 1 1 0 1 and S = 0 −1 1 0 . The following propositions hold.

Proposition_{1.17. For every z}_{∈ H there exists A ∈ Mod}₁_{such that A(z)}_{∈ D.} Two points z, z′ _{∈ D belong to the same orbit if and only if |z| = 1 and z}′₌_−1/z

or Re(z) =_{±1/2 and z}′_{= z}_{∓ 1.}

Proposition _{1.18. The stabilizer Stab(z) =} _{{A ∈ Mod}₁ _{: A(z) = z}_{} of a} point z_{∈ D is trivial except for three cases:}

(1) Stab(i) =_{{id, S}}

(2) Stab(e2πi/3) =id, ST, (ST )2 (3) Stab(eπi/3) =id, T S, (T S)2_.

Since the left quotient onH by PSL(2, Z) commutes with the right multiplica-tion by et/2 ₀ 0 e−t/2 ,

the geodesic flow on T(1)_{H descends to a flow - also called geodesic - on T}(1)

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7. FUNDAMENTAL DOMAIN 11

7. Fundamental domain

We would like to find a fundamental domain inH for the action of the geodesic flow on_M1. This amounts to specify some ”canonical” coordinates for every torus.

Our choice will be somewhat different from the usual ones. Any given torus is the quotient ofC by some lattice L = Zv1⊕Zv2with v1∧v2= 1. Let v1= A+iB, v2=

Ci + D.

First, we treat tori that lead to trivial geodesics in_M1. Suppose the lattice L

contains a vertical vector (0, d); then the associated geodesic is a curve of the form t_7→ a_{· e}t/2 _b · e−t/2 0 d· e−t/2 ≈ b_d+ ieta d.

Thus, the geodesic is a vertical halfline and its dynamics inM1 is trivial.

If L contains a horizontal vector, say (c, 0), the situation is analogous, for in this case t7→ a· et/2 _b_{· e}−t/2 c_{· e}t/2 ₀ ≈ a_c − ie−tb c.

Then, in the sequel, we shall restrict ourselves to lattices that do not contain vertical or horizontal vectors. Tori whose lattice contains either a horizontal or a vertical vector will be called degenerate.

Choose a vector w2= (x, y)∈ R2≈ C of the lattice L = Zv1⊕ Zv2 such that

|y| = min {|y′| : (x′, y′)_{∈ L and 0 6 |x| < 1)} .}

Notice that this set is non-empty: indeed, if Am + Cn = 0 for some m, n_{∈ N, then} it contains (0, y); otherwise Am + Cn is arbitrarily small as m and n vary, and we are done as well. Suppose that y > 0. By definition w2 = γv1+ δv2, for some

γ, δ∈ Z. It follows that (γ, δ) = 1, for otherwise y would not be the minimum. Then, there exists (α′, β′)∈ Z2_{such that α}′_δ_{− β}′_{γ = 1: this pair is not unique}

at all, but every other solution belongs to the set R ={(α′_{+ pγ, β}′_{+ pδ) : p}_{∈ Z}.}

Choose (α, β) in this set such that αv1+ βv2= (u, w) satisfies

u = min_{u′>1 : (u′, v′) = α′v1+ β′v2 with (α′, β′)∈ R} ,

and then take w1 = αv1+ βv2. If in this construction y < 0, just choose α, β

satisfying αδ− βγ = −1 and reverse the order of w1and w2.

To represent the torus, for the moment we use the matrix whose rows are the coordinates of w1and w2. We summarize the result of the procedure just described.

Proposition _{1.19. Any non-degenerate torus of area 1 can be represented as} M =

a b c d

, where M belongs to either Ω0 or Ω1, where

Ω0= a b c d ∈ SL(2, R) : 0 < c < 1 6 a, 0 < −b < d ; Ω1= a b c d ∈ SL(2, R) : 0 < a < 1 6 c, 0 < d < −b . At this point we restrict ourselves to matrices

a b c d

∈ ˜Ω = Ω0∪ Ω1 such

that a/c, b/d /_{∈ Q: this ensures that the operations we are going to carry out in the} next sections are always well defined. This restriction is not so strict, since we are neglecting a set of zero Lebesgue measure.

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8. Representation of the geodesic flow

The geodesic flow on _M1 can be written locally as right multiplication byTt

of a matrix A belonging to Ω = Ω0∪ Ω1. Eventually A· Tt gets out of Ω, but it

can be brought back to Ω by applying an element of Mod1. So, we should study

more deeply the identifications between borders of Ωi (i = 0, 1).

Consider

Σ0= Ω0∩ {a = 1} and Λ0= Ω0∩ {c = 1} ;

Σ1= Ω1∩ {c = 1} and Λ1= Ω1∩ {a = 1} .

The sets Λ1and Σ0 are naturally identified: take

1 b c d ∈ Λ1 and observe that 1 0 −[c] 1 1 b c d = 1 b {c} d − [c]b ;

hence, we are left to check that the matrix of the right-hand side belongs to Ω0:

indeed, 0 6 _{{c} < 1 and d − [c]b > −b, clearly (as [c] > 1). Let φ}0 : Λ1 → Σ0 be

the map such that φ0: 1 b c d 7→ 1 b {c} d − [c]b . There is a similar identification between Λ0 and Σ1, namely

1 _−[a] 0 1 a b 1 d = {a} b − [a]d 1 d . Let φ1: Λ0→ Σ1 be defined by φ1: a b 1 d 7→ {a} b − [a]d 1 d .

Proposition _{1.20. The first return map of the geodesic flow on ˜}_{Σ = Σ}₀_{∪ Σ}₁ is conjugated to the map

G : ˜Σ_{→ ˜}Σ, _{G(x, y, ε) =} 1 x , x2_{y + (} −1)ε_{x, 1} − ε . Proof. _{Notice that Σ}₀ _{is parametrized by (c, b) with (c, b, 0)}_{∈ P}₀_:

P0= (x, y, 0) : x_{∈ [0, 1], y ∈} −_{1 + c}1 , 0 . Analogously, Σ1 is parametrized by (a, d), with (a, b, 1)∈ P1:

P1= (x, y, 1) : x_{∈ [0, 1], y ∈} 0, 1 1 + a . Define Φ :_P0∪ P1→ ˜Σ by Φ(x, y, 0) = 1 y x xy + 1 and Φ(x, y, 1) = x xy− 1 1 y . Given 1 b c d

∈ Σ0, let the flow act for a time tR =−2 log c, to get into Λ0.

At this point, use the identification between Λ0 and Σ1 explained in the previous

paragraph: Z = φ1◦ TtR: 1 b c d 7→ 1/c bc 1 cd 7→ {1/c} bc − [1/c] cd 1 cd .

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8. REPRESENTATION OF THE GEODESIC FLOW 13 If a b 1 d

∈ Σ1 we have tR=−2 log a and

Z = φ0◦ TtR: a b 1 d 7→ 1 ab 1/a ad 7→ 1 ab {1/a} ad − [1/a]ab . In any case, Φ−1◦ Z ◦ Φ = G,

since d = bc + 1 (if a = 1), and b = ad_{− 1 (if c = 1).} More precisely, every M ∈ Ω0∪ Ω1 can be written uniquely asTt(x, y, ε), with

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CHAPTER 2

Interval exchange transformations

In this and in the following chapter we introduce the notions of interval ex-change transformation and translation surface. Motivations for these new defi-nitions come from the first chapter: we studied tori realized as suspensions over rotations, where the holomorphic one-form dz specified the horizontal and vertical directions.

A natural generalization of this object is thus a pair (X, ω) (a translation surface), where X is a Riemann surface of genus g and ω is a holomorphic one-form defined on it. Moreover, we shall require X to be realized as a suspension over an interval map of particular type (that generalizes the case of rotations). In the first section we define these more complicate ”rotations”, called interval exchange transformations (IET). The action of an IET is as follows: it breaks the interval of definition into a finite number of subintervals and rearranges them in another order. It is clear that an IET is thus specified by the lengths of the subintervals and by the order of them before and after rearranging.

It turns out there are some natural operators in the space of IETs, called Rauzy-Veech and Zorich induction maps ( ˆR and ˆZ) that ”generalize” the Euclidean algorithm (respectively, the slow algorithm and the normal one). Given an IET T : I → I, ˆR is simply the first return map of T on a convenient subinterval and ˆZ is its acceleration: ˆZ(T ) = ˆRn_{(T ), for some n depending on T . The three IETs T ,}

ˆ

R(T ) and ˆZ(T ) have the same number of subintervals. We will study in detail the dynamics of these operators.

In the second part of the chapter, we will focus on combinatorial properties of IETs. There are some operators, called reduction and extension, connecting interval exchange transformations with a different number of subintervals. To properly study these operators, we will also introduce some special classes of permutations and spend some time explaining the mutual relations among them.

1. Definitions

Roughly speaking, an interval exchange transformation (IET) is a map T : I→ I defined on a bounded interval I, say I = [0, λ∗_{), which breaks I into subintervals}

Ij= [aj, bj), 1 6 j 6 d and rearranges them in a different order.

For the formal definition, we will adopt this point of view: we give an invariable name to every subinterval and specify its place before and after rearranging by a pair of permutations. More precisely, any IET T is determined by (π, λ):

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• λ = (λα)α∈A∈ RA+, where A is a finite set (called the alphabet),

• π = (π0, π1), π0, π1:A → {1, . . . , d} permutations, where d = |A|;

then I = [0, λ∗) with λ∗=P

α∈Aλα, and I is the union of the d intervals of length

λαin the order specified by π0, while T (I) is the union of the same intervals in the

order specified by π1. Hence, T (x) = x + δα on Iα, where the translation vector

(δα)α∈A is given by δα= X π1(β)<π1(α) λβ− X π0(β)<π0(α) λβ.

We may write δ = Ω(λ), where Ω is the linear operator whose coefficients (Ωα,β)α,β∈A

(in the canonical basis of RA_{) are given by the formula}

Ωα,β =    1 if π0(α) > π1(α) and π0(β) < π1(β) −1 if π0(α) < π1(α) and π0(β) > π1(β) 0 otherwise.

Sometimes we write Ωπ instead of Ω, to specify that Ωπ is determined by the pair

of permutation π = (π0, π1).

In the sequel we will use_{|λ| to indicate the sum}P

α∈Aλα.

1.1. Symplectic form. Observe that Ωπis skew-symmetric. It is well-known

that a skew-symmetric operator L : Rd

→ Rd _{is not always an isomorphism: its}

eigenvalues are complex numbers with zero real part. In particular, if d is odd, dim ker L > 1. We can define an alternate bilinear form ˆωπ:RA× RA→ R setting

ˆ

ωπ(v, w) =−hv, Ωπ(w)i.

If Ωπ is not an isomorphism, this form is degenerate: ˆωπ(·, w) ≡ 0 if w ∈ ker Ωπ.

However, to make it non-degenerate (and hence symplectic, by definition) we can restrict the domain to Hπ= Ωπ(RA):

ωπ(Ωπ(v), Ωπ(w)) =−hv, Ωπ(w)i.

This works because ker L = L Rd⊥

, for any skew-symmetric L. As pointed out before, dim Hπ is an even number: we will write dim Hπ = 2g(π) or sometimes

dim Hπ = 2g.

1.2. Irreducible permutations. In what follows we shall restrict our atten-tion to irreducible pairs of permutaatten-tions: π = (π0, π1) is called irreducible if

π1◦ π0−1({1, . . . , k}) = {1, . . . , k} implies k = d.

If π is reducible, then T = (π, λ) splits into two IETs (defined on disjoint invariant subintervals of I) with simpler combinatorics: this is the reason for considering only irreducible permutations. Given an alphabet A, SA will denote the set of

irreducible pairs of permutations of the given alphabet, while SA denotes the full

set of pairs.

2. Rauzy-Veech and Zorich maps

Let T : I _{→ I be an IET with data (π, λ). Assume that I = [0, 1). We want to} describe a procedure to associate to T another IET defined on a smaller interval. This is the analogous of the ”slow Euclid’s algorithm” in the case of two intervals: (λ, λ′)7→ (min {λ, λ′_{} , max {λ, λ}′_{} − min {λ, λ}′_}).

Let α(0) = π−10 (d) be the name of the rightmost subinterval of I, and α(1) =

π1−1(d) the name of the rightmost subinterval of T (I). Consider the set ∆1=SA×

λ ∈ RA

+ : λα(0)6= λα(1) . Thus, for data in ∆1, λα(0)− λα(1)6= 0: if λα(0)> λα(1)

we say that T is of type 0 (or top type), if λα(0)< λα(1), T is of type 1 (or bottom

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2. RAUZY-VEECH AND ZORICH MAPS 17

either case, the winner is the symbol that corresponds to the larger interval. Then, ˆ

R : ∆1 → SA× RA+, called the Rauzy-Veech induction map, acts as follows. The

image ˆR ((π, λ)) = (π′, λ′) satisfies (ε is the type of (π, λ)) • λ′ α(ε)= λα(ε)− λα(1−ε) and λ′β = λβ if β6= α(ε); • πε= π′εand π1−ε′ (β) =    π1−ε(β) if π1−ε(β) < π1−ε(α(ε)); π1−ε(β) + 1 if π1−ε(α(ε)) 6 π1−ε(β) < d; π1−ε(α(ε)) if π1−ε(β) = d.

Checking that π′ _{is an irreducible pair is a trivial matter. So ˆ}_{R preserves the}

irreducibility.

We may write λ′_{= V λ, where V = I}_{− E}

α(ε),α(1−ε). Here I is the identity d× d

matrix and Ei,j is the elementary matrix whose entries are zero, except for eij = 1.

Note that V−1= I + Eα(ε),α(1−ε).

Let us compare the translation vectors δ and δ′ of (π, λ) and (π′, λ′). Easily, we get

δα(1−ε)′ = δα(ε)+ δα(1−ε) and δ′β= δβ otherwise.

Thus, δ′_{= Θδ where Θ = Θ}

π,λ= I + Eα(1−ε),α(ε). Observe that V = Θ−1∗, and we

prefer to work with Θ rather than V since the former has nonnegative coefficients. Notice also that Θπ,λ depends just on π and λ/|λ|.

Lemma _{2.1. If ˆ}_{R(π, λ) = (π}′_{, λ}′_{), then Θ}_π,λ_Ω_π _{= Ω}_π′V_π,λ and Θ_π,λΩ_πΘ∗

π,λ =

Ωπ′.

Proof. _{By definition, V (λ) = λ}′_{, Θ}_π_{(λ) = δ, Θ}_π,λ_{(δ) = δ}′_{, Ω}_π′(λ′) = δ′ and

V−1 = Θ∗.

We would like to iterate the Rauzy map ˆR infinitely many times: let ∆ndenote

the set where ˆRn _{is defined. Thus we are interested in the intersection ∆} ∞ =

T

n∈N∆n: no iterate of (π, λ) ∈ ∆∞ is such that λε = λ1−ε. Clearly, if the λα’s

are rationally indipendent, then (π, λ) _{∈ ∆}∞: it follows that ∆∞ is a set of full

Lebesgue measure in RA +.

Observe that if ˆRn_{(π, λ) = (π}n_{, λ}n_{), then}

λn= Θ−n∗_π,λ(λ), where (2.1) Θ−n∗π,λ = Θ −(n−1)∗ πn−1_,λn−1· · · Θ−1∗_π,λ. .

Let us introduce now the Zorich induction map. In the case of two intervals, it corresponds to the usual Euclid’s algorithm:

(λ, λ′)_7→l, w₋hw l

i l,

where w = max_{{λ, λ}′_{}, l = min {λ, λ}′_{} and [ · ] denotes the integer part.}

For (π, λ)∈ ∆∞, set (πn, λn) = ˆRn(π, λ). Then ˆZ ((π, λ)) = ˆRn(π,λ)((π, λ)), where

n = n(π, λ) is the minimum n∈ N such that type(π, λ) 6= type(πn_{, λ}n_{) (obviously,}

such a number always exists).

The Rauzy-Veech and Zorich renormalization maps are the projectivizations of the corresponding induction maps. More precisely, suppose the initial interval has length one, i.e. _{|λ| ≡}P

α∈Aλα= 1. Then, (π′, λ′) = ˆR(π, λ) is such that

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since we have subtracted from_{|λ| the length of the loser. We define the Rauzy-Veech} renormalization map R as R(π, λ) = π′, λ ′ 1− λα(1−ε) , so that the new IET is defined again on a unit length interval.

The Zorich renormalization map is defined similarly: Z(π, λ) = Rn(π,λ)(π, λ),

where as before n(π, λ) is the minimum n such that type(Rn_{(π, λ)) = 1}_{−type(π, λ).}

3. Dynamics of the induction map

In this section we prove a few elementary facts about induction maps. Let (πn_{, λ}n_{) = R}n_{(π, λ). The first trivial observation is that the type ε}n _{does change}

infinitely many times. Let wn and ln be the symbols corresponding to the winner

and the loser of (πn_{, λ}n_{), respectively: w}

n = α(εn) and ln = α(1− εn). The next

proposition shows that the dynamics of R is quite rich.

Proposition _{2.2. Both sequences (w}_n₎_n∈N _{and (l}_n₎_n∈N _{assume each β} _{∈ A} infinitely many times.

Proof. _{If we prove the claim for w}_n_{, then the claim for l}_n _{follows, because} every winner becomes a loser after a finite number of iterations. Let _{B 6= A be the} set of symbols β that appear only finitely many times in (wn)n∈N: replacing (π, λ)

with (πN_{, λ}N_{) we may suppose that they do not appear at all. Since they never}

win, their length does not change: λn

β = λβ, for every n > 1.

We claim that each β _{∈ B appears only finitely many times in (l}n)n∈N. If not,

take γ _{∈ B such that γ = l}n for infinitely many n: there exists a symbol α such

that (α, γ) = (wn, ln) for infinitely many n∈ N: the fact that λγ is constant implies

that λn

αwould become eventually negative. This justifies our claim.

So, up to replacing (π, λ) with some iterate, we suppose that no β∈ B occurs in (ln)n∈N: for any β∈ B and ε ∈ {0, 1}, there does not exist any n ∈ N such that

πn

ε(β) = d.

Fix β ∈ B and let mβ

ε = maxn∈N{πεn(β)} < d. When β reaches its maximum

position mβ

ε, then it cannot decrease (because, to decrease, it should first become

a loser): this shows that the sequence πn

ε(β) is eventually constant. Then, up to

replacing (π, λ) with (πN_{, λ}N_{) (where N > max}

β∈Bmaxεmβε), we may assume that

the two sequences (πn

0(β))n∈N and (π1n(β))n∈N are constant.

To conclude, we will prove that this contradicts irreducibility. Indeed, take α_{∈ A\B, β ∈ B and suppose π}ε(α) < πε(β): this implies that α will be always to

the left of β. Since α /_{∈ B, it will be a winner of π}n _(w

n = α(εn) = α) for some

n > 1 and εn_{= 1}

− ε, clearly: this contradicts the fact that πε(β) is constant, since

πn = · · · α · α · β · · 7→ πn+1₌ · · · α · α · · β · .

As this holds for every α /∈ B and β ∈ B, we obtain that πε(α) > πε(β). This

contradicts irreducibility.

As a corollary, we prove that the matrices Θ∗n _{become ”complicate”, as n is}

large.

Corollary _{2.3. For each m}_{∈ N, there exists n ∈ N such that all the entries} of the matrix Θ∗nπm_,λm are positive.

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4. RAUZY CLASSES AND RAUZY DIAGRAMS 19

Proof. _{To simplify the notation, let Θ}∗_{(α, β, m, n) be the (α, β) entry of the} matrix Θ∗n

πm_,λm. Recall the definition of Θ∗_π,λ: Θ∗(α, β, m, 1) = 1 if either (α, β) =

(wm_{, l}m_{) or α = β and Θ}∗_{(α, β, m, 1) = 0 otherwise.}

Next, observe that, in general (use (2.1)), (2.2) Θ∗(α, β, m, k + l) =X γ Θ∗(α, γ, m, k)Θ∗(γ, β, m + k, l) and so Θ∗(α, β, m, n + 1) =X γ Θ∗(α, γ, m, n)Θ∗(γ, β, m + n, 1) >Θ∗(α, β, m, n)Θ∗(β, β, m + n, 1) = Θ∗(α, β, m, n);

that is, Θ∗(α, β, m, n + 1) > Θ∗(α, β, m, n), for every n∈ N. Given α ∈ A we prove that there are an enumeration γ1, . . . , γd ofA and a sequence of natural numbers

n1, . . . , nd such that

Θ∗(α, γi, m, ni) > 0 for all i∈ {1, . . . , d} :

this will prove the corollary.

Let γ1= α and n1= 1. Clearly, Θ∗(α, γ1, m, n1) > 1 > 0.

Using the previous proposition, γ1 will be the winner of πm2, for some m2 > m.

Then, set γ2= lm2: γ2is the loser of π

m2_{. By definition, we have Θ}∗_(γ

1, γ2, m2, 1) =

1 and so, using (2.2) with k = m2− m and l = 1 we obtain that

Θ∗(α, γ2, m, m2− m + 1) > Θ∗(α, γ1, m, m2− m)Θ∗(γ1, γ2, m2, 1) > 1,

since α = γ1. Now set n2 = m2− m + 1: if d = 2 we are done, otherwise we

have to find γ3 and n3. Let us explain only the next step: the general idea will be

sufficiently clear.

By the previous proposition, there is p2> m2 such that wp2 is neither γ1 nor

γ2; moreover, there is m′ > p2 such that either wm

′

= γ1 or wm

′

= γ2. Take the

minimum such m′ and call it m3; let γ3 = lm3. Of course lm3 = wm3−1, since

wm3−1_{∈ {γ}_/

1, γ2}: in particular lm3 is neither γ1nor γ2. Also, Θ∗(γj, γ3, m3, 1) = 1

(where γj = lm3) and so

Θ∗(α, γ3, m, n) > Θ∗(α, γj, m, m3− m)Θ∗(γj, γ3, m3, n− m3+ m)

>Θ∗(α, γj, m, m3− m)Θ∗(γj, γ3, m3, 1) > 1,

as long as n > m3− m. We are done, setting n3= m3− m + 1. The general case

can be treated in the same way.

4. Rauzy classes and Rauzy diagrams

We focus our attention on how inductions act on permutations. Suppose that ˆ

R(π, λ) = (π′_{, λ}′_{). If λ}

α(0) > λα(1), then π′ is obtained from π applying a top

operation, that is: t : π = · · · β · · β γ · α 7−→ π′ = · · · β · · β α γ · ;

in other words, the whole top row and the symbols to the left of the winner (included the winner) in the bottom row do not change, while the symbols in the bottom row to the right of the winner are rotated one place to the right. The bottom operation is similar: if λα(0)< λα(1) the transformation rule is

b : π = · · α β · γ · · · α 7−→ π′₌ · · α γ β · · · · α ;

the bottom row remains unchanged and the symbols in the other line to the right of the winner are rotated to the right. So, we may define a diagram as follows. Take

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the irreducible pairs of permutations _SAto be the set of vertices. Connect π with

π′ _{by an t-arrow (an arrow labeled by t, or top arrow) if π}′ _{is obtained applying}

a top operation to π. In the same way we define b-arrows. The resulting oriented graph is called Rauzy diagram. As pointed out before, top and bottom operations preserve the irriducibility of pairs; moreover, they are clearly invertible (for any arrow t, tk

◦ t = id for some k < d, and the same for b). So, t, b : SA → SA are

bijections.

Paths inSAare defined concatenating t-arrows and b-arrows. Let Π(A) denote

the set of paths inSA: the set of irreducible pairsSA is naturaly decomposed into

connected components, called Rauzy classes. Given π ∈ SA, let R(π) denote the

Rauzy class containg π. For any π, π′∈ R, there exists a path contained in R going from π to π′ (and also another one going from π′ to π). The set of path starting and ending at π will be denoted Π(π) and called Rauzy monoid. When we speak of the action of the Rauzy monoid Π(π) of Hπ, we mean the action obtained via

Θ. Precisely, recall that Θπ,λ depends only on π and on the type of λ; thus any

arrow σ determines Θπ,λ, where π is the start of σ and λ is any length vector whose

type coincides with the type of σ. A path thus gives a sequence of operators: we compose them according to concatenation. As a notation, σ0σ1represents the path

following σ0 and then σ1: in this case

Θ(σ0σ1) = Θ(σ1)◦ Θ(σ0).

Observation _{2.4. Notice that the two leftmost symbols of any π are preserved} by top and bottom operations. Thus, they are constant on R(π).

5. Combinatorial tricks

There are a couple of combinatorial operations we can do with IETs. In spite of their simplicity, they will play a crucial role in what follows.

5.1. Reduction. Let π = (π0, π1)∈ SA, where d =|A| > 3.

Definition _{2.5. We call π}′ _{= (π}′

0, π1′) a reduction of π if it is irreducible and

it is obtained from π by deleting one symbol ζ ∈ A in π0 and π1.

Set _A′ ₌_{A\ {ζ}, H} π′ = Ω_π′(RA ′ ), dim Hπ′ = 2g(π′). Let P :RA → RA ′ be the projection: then π′_{∈ S}

A′ by definition and P Ω_πP∗ = Ω_π′.

We must understand how the genus changes after a reduction. There is a simple answer.

Lemma _{2.6. Let π}′ _{be a reduction of π (ζ denotes the deleted symbol). There} are two possibilities: g(π) = g(π′) or g(π) = g(π′) + 1. Moreover the first case is equivalent to any of the following:

(1) eζ ∈ H/ π; (2) Hπ= Ωπ(RA ′ ); (3) eζ ∈ Ω/ π(RA ′ ); (4) P|Hπ: Hπ→ Hπ′ is a symplectic isomorphism.

Proof. _{Note that Ω}_π_(e_α_{) = 0 implies that rank P Ω}_π_P∗_{= rank Ω}_π_{, i.e. g(π) =} g(π′_{). Moreover, Ω}

π(eα) 6= 0 implies that 2g − 2 6 rank Ω(π′) 6 2g− 1, i.e.

g(π′_{) = g(π)} _{− 1, since rank Ω(π}′_{) is even. Thus g(π) = g(π}′_{) if and only if}

Ωπ(eα) = 0, and this is easily shown to be equivalent to all the other statements.

Corollary _{2.7. If g(π) = g(π}′_{) + 1, then P restricts to a symplectic} isomor-phism P : H[eζ]

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5. COMBINATORIAL TRICKS 21

Proof. _{Let us characterize elements in (H}_π₎

[eζ]. Given h∈ H = Hπ, we can write h =P ν∈Acν(Ωπ(eν)), cν ∈ R. Then ωπ(h, eζ) = X ν∈A cνheν, eζi = cζ, so h _{∈ H}[eζ] if and only if h∈ ΩπP ∗₍_RA′

) (P∗ _{is the inclusion). Recalling that}

P ΩπP∗ = Ωπ′ we get that P : H_[e

ζ] → Hπ′. Since P (eζ) = 0, P is defined also

as P : H[eζ] → H

π′: in this case it is a symplectic isomorphism. That it is an

isomorphism is obvious, moreover, for µ, ν _{∈ A}′_,

ωπ′(P Ω_π(e_µ), P Ω_π(e_ν)) = ω_π′(Ω_π′(e_µ), Ω_π′(e_ν)) =

−heµ, Ωπ′(e_ν)i = −he_µ, Ω_π(e_ν)i = ω_π′(Ω_π(e_µ), Ω_π(e_ν)),

so it is symplectic, too.

5.2. Extension. Let _|A′_{| 6 2 and π}′ _{∈ S}′

A. We want to get a pair π that

reducts to π′_{. To this aim, let ζ be a symbol that does not belong to the alphabet}

A′_{, define}_{A = A}′_{∪ {ζ} and let α and β are the two leftmost symbols in the rows}

of π′_{. Choose an ordered pair (γ, δ)} _{∈ A}′_{× A}′_{\ {α, β} and finally define π to be}

the pair of permutations obtained from π′ _{by inserting ζ in the top row just before}

γ and in the bottom row just before δ. π′ = α · γ · · · β · · · δ · 7→ π = α · ζ γ · · · · β · · · · ζ δ · . This definition is useful because it makes the following lemma hold.

Lemma _{2.8. An extension π of a irreducible permutation π}′ _{is also irreducible.}

Proof. _Obvious.

By Observation 2.4, we may speak of the extension E = Eα,β,γ,δ of the whole

Rauzy class R(π′). Hence, in the remaining part of this section we fixE (indeed, it means that we are fixing γ, δ, since α and β are determined by π′).

Lemma _{2.9. If π}′′ _{∈ R(π}′_{), then} _E(π′′₎_{∈ R(π). In other words, E(R(π}′₎₎_⊂ R_(π).

Proof. _{Since Rauzy classes are connected, it suffices to prove that} _E(b(π′₎₎ and _E(t(π′_{)) belong to R(π). There are three different cases:}

(1) π′ = α · γ · · η β · · η · δ 7→ π = α · ζ γ · · η β · · η · ζ δ (2) π′ ₌ α _{· η · · γ} β _{· · δ · η} 7→ π = α _{· η ·} _{· ζ γ} β _{· · ζ δ · η} (3) π′ ₌ α _{· γ · · ·} β _{· · · δ ·} 7→ π = α _{· ζ γ ·} _{· ·} β _{· ·} _{· ζ δ ·} .

The third case is the easiest: in fact, t(π) = _E(t(π′_{)) and b(π) =} _E(b(π′_{)). In}

the second case, b2_{(π) =}

E(b(π′_{)) and t(π) =}_E(t(π′_{)); the first case is analogous}

(exchange b with t).

This lemma allows us to extend the definition of _{E to the space of paths of} R_(π′_{) (with values in the space of paths in R(π)):}

E∗: Π (R(π′))→ Π (R(π)) .

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5.3. Extension and reduction. Let us relate extension with reduction. The first obvious observation is the following.

Lemma _{2.10. If π is an extension of π}′_{, then π}′ _{is a reduction of π. The} converse holds if and only if the deleted letter is not the rightmost symbol in either row of π.

To fix the notation, let π = E(π′_{) (ζ denotes the added symbol). Then}_{A =}

A′_{∪ {ζ}, P : R}A_{→ R}A′

is the projection (i.e. P (eζ) = 0 and P acts as the identity

onRA′

⊂ RA_).

Lemma _{2.11. Let σ}′_{∈ Π(R(π}′_{)) be a path and σ =}_E(σ′_{) its extension. Then} P Θ(σ) = Θ(σ′)P .

Proof. _{It suffices to prove the claim when σ}′ _{is an arrow. Recall the three} cases in Lemma 2.9. We shall spell out the details only for the first case (with respect to a top arrow), the other cases being analogous.

By definition, Θ(σ′₎_{· e}

η = eη+ eδ and Θ(σ′)· eν = eν if ν ∈ A′\ {η}; in the same

way, since t2_{(π) =}

E(t(π′_{)), Θ(σ)}_·e

η = eη+ eδ+ eζ and Θ(σ)·eν = eνif ν∈ A\ {η}.

Thus the claimed equality follows.

Corollary _{2.12. If g(π) = g(π}′_{), P is a symplectic isomorphism P : H}_π _→ Hπ′ that conjugates Θ(σ)|H_π and Θ(σ′)|H_π′.

Lemma_{2.13. Every π}_{∈ S}_A_with_{|A| > 2 can be realized as an extension: there} exist ζ _{∈ A and π}′_{∈ S}

A\{ζ} such that π is an extension of π′.

Proof. _{We distinguish four cases:} (1) π = α _{· ·} _· β _· β _{· · α · ·} (2) π = α · · β · · β · · · α · (3) π = α _{· · β · ·} β _{· · α · ·} (4) π = α _{· · · · β} β _{· · · · α} corresponding to π1(α) < π0(β), π1(α) > π0(β), π1(α) = π0(β) < |A|, π1(α) =

π0(β) =|A|. Let, respectively, ζ = α, ζ = β, ζ = α, ζ ∈ A\ {α, β}. Define π′ to

be the reduction of π obtained deleting ζ. It is trivial to check that indeed π′ _is

irreducible, and that π is an extension of π′_.

6. Standard, good and degenerate permutations

We go on studying further combinatoric properties of irreducible pairs of per-mutations.

Definition _{2.14. A pair π = (π}₀_{, π}₁_{) is called standard if the first symbol of} each line is the last symbol of the other line. Equivalently, π is standard if and only if π₀−1(1) = π−1₁ (d) and π₀−1(d) = π−1₁ (1), where d =_{|A|, as usual. Graphically,}

π = α _{· · · · β} β · · · · α .

Every standard pair π is of course irreducible; moreover the matrix associated to Ωπ with respect to the basis of RA ordered according to π0 is of the following

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6. STANDARD, GOOD AND DEGENERATE PERMUTATIONS 23

form (just use the definition of Ωπ):

(2.3)       0 1 _{· · · 1} −1 . .. ... .. . . .. 1 −1 · · · −1 0       .

Lemma _{2.15. Every Rauzy class contains a standard pair (of permutations).} Proof. _Let π = α · · β · · β · · · · · .

If β 6= π0(d), we can find, by Sub-lemma 2.16 below, some π′∈ R(π) such that

π′= α · · · · β β · · · · · .

The top operation applied to π′_{leaves the top line of π}′_{fixed and rotates the symbols}

in the bottom line at the right of β. Thus, after at most d_{− 2 top operations, we}

obtain a standard pair.

Sub-lemma _{2.16. Let π be an irreducible permutation. Given ζ} _{6= π}−1

0 (1) (ζ

is not the first symbol in the top line), there exists π′ ∈ R(π) such that π′

0(ζ) = d

(ζ is the rightmost symbol in the top line of π′_).

Proof. _{Recall that the two leftmost symbols are constant in each Rauzy class.} Let A0 _{be the set of symbols that do not reach the rightmost position in any}

π′_{∈ R(π):}

A0₌

{γ ∈ A : π′0(γ) < d for every π′∈ R(π)} .

Let k be the maximum value of π′0(γ) over π′ ∈ R(π) and γ ∈ A0. We want to

prove that k = 1, so that the unique symbol in_A0_{is π}−1

0 (1). This would imply the

conclusion.

Let αk ∈ A0be a symbol for which the maximum is attained: π0′(αk) = k for some

π′. No bottom operation can move αk (because it would move αk to the right), so

the position of αk is constant on the whole Rauzy class. With the same argument,

the symbols to the left of αk are constant, too. Let as call them α1, . . . , αk−1.

Now, repeat the same steps for the bottom line, defining A1_{, h (the analogous of}

k) and β1, . . . , βh (the analogous of the αj’s).

Every pair in R(π) is of this form:

α1 α2 · · · αk · · ·

β1 β2 · · · βh · · ·

.

In particular, α1, . . . , αk−1all belong toA0and cannot be in the rightmost position

in the bottom line (otherwise a sequence of bottom operations would take αkin the

last position in the top row). Then {α1, . . . , αk−1} ∈ A1. Symmetric statements

hold for β1, . . . , βh. Therefore,

{α1, . . . , αk−1} ⊂ {β1, . . . , βh} and {β1, . . . , βh−1} ⊂ {α1, . . . , αk}

and k = h_{− 1, k = h + 1 or k = h. The first two cases contradict irreducibility at} once. So we are left that k = h implies k = 1. If _{α1, . . . , αk−1} = {β1, . . . , βh−1}

then again irreducibility is violated, hence we may suppose that βh= αj for some

j 6 k_{−1. But this is also impossible, since in this case {α}1, . . . , αk} = {β1, . . . , βh}.

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Let π be a standard permutation, say π = α _{· · · · β} β · · · · α .

There are two important cases to distinguish. We call π degenerate if either π0−1(2) = π1−1(2) or π−10 (d− 1) = π−11 (d− 1), that is π = α ζ _{· · · β} β ζ _{· · · α} or π = α _{· · · ζ β} β _{· · · ζ α} .

We call π good if its double reduction obtained forgetting α and β (i.e. the two rightmost symbols) is an irreducible pair.

Lemma _{2.17. Every Rauzy class (}_{|A| > 3) contains either a degenerate or a} good pair.

Proof. _{Assume, by contradiction, that the claim is false for some Rauzy class} R_{. Let π}_{∈ R be a standard permutation (use Lemma 2.15). As usual, we put}

π = α · · · · β β _{· · · · α} .

The pair π′_{obtained deleting α and β is irreducible, that is there exists 1 6 k 6 d}₋₃

such that the first k symbols in top row of π′ _{coincide with those in the bottom}

row. Since π is non-degenerate, 2 6 k 6 d_{− 4. Assume we have chosen π among} other standard pairs in R so that the maximum k is attained. Let us call γ1, . . . , γk

and δ1, . . . , δk the first k symbols in the top and bottom row of π′, respectively.

Then, π looks like

π = α γ1 · · · γk · · β β δ1 · · · δk · · α , with {γ1, . . . , γk} = {δ1, . . . , δk} and γ16= δ1.

Define now ζ = π−10 (d− 1) and l = π1(ζ) its position in the bottom line. Clearly,

l > k + 1 and l 6 d− 2 (by non-degeneracy of π). There are two possibilities: (1) l = k + 1, π = α γ1 · · · γk · · ζ β β δ1 · · · δk ζ · · α ; (2) k + 1 < l 6 d_{− 2,} π = α γ1 · · · γk · · ζ β β δ1 · · · δk · ζ · α .

Now, let η be the symbol preceding ζ in the bottom line, i.e. η = π−11 (l− 1) and

let p = π0(η) be its position in the top row. In case (1), η = δk and 2 6 p 6 k + 1.

In case (2), k + 2 6 p 6 d_{− 2. Graphically,} (1) π = α γ1 · · · γp−1 = η · · · γk · · ζ β β δ1 · · · δk= η ζ · · α ; (2) π = α γ1 · · · γk · η · · ζ β β δ1 · · · δk · · η ζ · α .

The stategy of the proof consists of obtaining from π (through top and row op-erations) another standard pair ˜π _{∈ R that violates the maximality of k or the} hypothesis.

The sequence of top and bottom operation is the following:

• apply d − p bottom operations: η becomes the rightmost symbol in the top row;

• apply d−l top operations: ζ becomes the rightmost symbol in the bottom row;

• apply p − 1 bottom operations: β becomes the rightmost symbol in the top row;

Geodesic flows on moduli spaces, continued fractions and Lyapunov exponents

Matematica

Anno Accademico 2006 – 2007

Tesi di Laurea Specialistica:

Geodesic flows on moduli spaces,

continued fractions and Lyapunov

exponents

Candidato:

Dario Beraldo

Relatore:

Prof. Stefano Marmi

Controrelatore:

Prof. Marco Abate

Contents

Preface

Geodesic flow on the hyperbolic plane

Interval exchange transformations